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A novel (k,n) secret sharing scheme from

A new grayscale image encryption algorithm based on threshold secret sharing is proposed. The
scheme allows a secret image to be transformed into n shares, where any shares can be used to
reconstruct the secret image, while the knowledge of or fewer shares leaves no sufficient information
about the secret image and it becomes hard to decrypt the transmitted image. In the proposed scheme, the
pixels of the secret image are first permuted and then encrypted by using quadratic residues. In the final
stage, the encrypted image is shared into n shadow images using polynomials of Shamir scheme. The
proposed scheme is provably secure and the experimental results shows that the scheme performs well
while maintaining high levels of quality in the reconstructed image.

A novel (k,n) secret sharing scheme from

1.
International Journal of Network Security & Its Applications (IJNSA), Vol.6, No.4, July 2014
DOI : 10.5121/ijnsa.2014.6406 65
A NOVEL (K,N) SECRET SHARING SCHEME FROM
QUADRATIC RESIDUES FOR GRAYSCALE IMAGES
El-Tigani B. Abdelsatir, Sahar Salahaldeen, Hyam Omar and Afra Hashim
Department of Computer Science, Faculty of Mathematical Sciences
University of Khartoum, Sudan
ABSTRACT
A new grayscale image encryption algorithm based on ሺ݇, ݊ሻ threshold secret sharing is proposed. The
scheme allows a secret image to be transformed into n shares, where any ݇ ൑ ݊ shares can be used to
reconstruct the secret image, while the knowledge of ݇ െ 1 or fewer shares leaves no sufficient information
about the secret image and it becomes hard to decrypt the transmitted image. In the proposed scheme, the
pixels of the secret image are first permuted and then encrypted by using quadratic residues. In the final
stage, the encrypted image is shared into n shadow images using polynomials of Shamir scheme. The
proposed scheme is provably secure and the experimental results shows that the scheme performs well
while maintaining high levels of quality in the reconstructed image.
KEYWORDS
Secret Sharing, Quadratic Residues, Secret Image Sharing, Image Encryption, Image Coding
1. INTRODUCTION
The use of digital images to communicate has increasingly become a new way of communication
nowadays especially in military, medical and financial applications. Therefore, protecting digital
images containing sensitive or confidential information has become an important issue.
Encryption is a widely used technique to achieve information security goals by transforming data
into unreadable format. However, in comparison with textual information, image data is different
in properties, such as the high correlation of image pixels and high redundancy. Therefore, the
application of conventional methods of encryption in digital images is not recommended.
Moreover, secret image sharing eliminates weaknesses in pure image encryption. For example, if
the secret key in encryption is lost the entire secret image becomes invalid. Also, in situations
where media failure is more common, there is a risk if the secret image is held by only one
person, the secret data may be lost or modified. This weakness can be avoided in ሺ݇, ݊ሻ threshold
secret sharing schemes in which the image is encoded into n shadow images and only ݇ ൑ ݊ of
them can reconstruct the original image.
The concept of secret sharing scheme was originally proposed by Shamir [1] and Blakely [3]
independently in 1979 for textual information. The research of secret sharing in digital images
first appeared in 1994 by the work of Naor and Shamir [2] in which the human visual system is
used directly to recover the secret image and the scheme was computationally efficient.
In recent years, along with the prevalent advancements in image processing, secret image sharing
also has been in active research. A wide variety of methods have been developed for processing
grayscale images [9] [13] [10], and also for colour images [6], [7],[8]. While many proposals
follow the classical idea of Shamir secret sharing scheme with variations, in some schemes, the
size of the shared images is much bigger than the original one. Moreover, a common

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International Journal of Network Security & Its Applications (IJNSA), Vol.6, No.4, July 2014
66
disadvantage of existing schemes is that there is a great quality loss between the secret image and
the recovered one [4] [12].
In this paper, we extend ሺ2,2ሻ Chen-Chang method [5] to ሺ݇, ݊ሻ secret sharing threshold scheme
and as a result we propose a new scheme for secret image sharing from quadratic residues and
incorporated with Shamir secret sharing method [1]. The proposed scheme is provably secure and
prevents the secret image from being revealed by making use of quadratic residues to encrypt the
secret image and then the encrypted pixels are shared into ݊ image shares by Shamir secret
sharing method. The scheme provides an advantage to Chen-Chang method [5] in terms of
flexibility in the number of secret shares which is limited to only two shares. Moreover, the
experimental results show nearly identical quality of the reconstructed image compared with the
original one. Also, the size of the secret image and the image shares is the same as the size of the
original secret image. Additionally important, the new scheme provides a better security measures
to protect the secret image in comparison with [5] and [4].
The rest of this paper is organized as follows. In Section 2, we review the schemes of Shamir,
Naor-Shamir, Thien-Lin and Chen-Chang. In Section 3, we propose efficient ሺ݇, ݊ሻ threshold
method for secret image sharing. In Section 4, we give experimental results and finally in Section
5 we draw our conclusions.
2. RELATED WORK
2.1. Shamir Secret Sharing
We begin by introducing the original Shamir's scheme [1]. Suppose that we want to encode the
secret S into n shares ሺS1, S2, … , Snሻ and we wish that the secret data S cannot be revealed
without k or more shares. In Shamir secret sharing scheme the partition of the secret is done by
the following polynomial:
Fሺx୧ሻ ൌ y ൅ mଵx୧ ൅ mଶx୧
ଶ
൅ ‫ڮ‬ ൅ mሺ୩ିଵሻx୧
ሺ୩ିଵሻ
modሺpሻ, i ൌ 1,2, … , n (1)
where y is the share, Sଵ , p is a prime number and the cofficients of the k െ 1 degree polynomial
m୧ are chosen randomly and then the shares are evaluated as S1 ൌ Fሺ1ሻ, S2 ൌ Fሺ2ሻ, … , Sn ൌ
Fሺnሻ
Given any k pairs of the share pairs ሼሺi, Siሻሽ, i ൌ 1,2. . , n. we can obtain the coefficients ݉௜ of
F(x) by largrange interpolation as follows:
S ൌ
ሺെ1ሻሺ୩ିଵሻ
ሾ Fሺxଵሻ
ሺ୶మሻሺ୶యሻ…ሺ୶ౡሻ
ሺ୶భି୶మሻሺ୶భି୶యሻ…ሺ୶భି୶ౡሻ
൅ Fሺxଶሻ
ሺ୶భሻሺ୶యሻ…ሺ୶ౡሻ
ሺ୶మି୶భሻሺ୶మି୶యሻ…ሺ୶మି୶ౡሻ
൅ ‫ڮ‬ ൅
Fሺx୩ሻ
ሺ୶భሻሺ୶మሻ…ሺ୶ౡషభሻ
ሺ୶ౡି୶భሻሺ୶ౡି୶మሻ…ሺ୶ౡି୶ౡషభሻ
ሿ (2)
2.2. Naor-Shamir Scheme
In 1994 Naor and Shamir [2] proposed the first secret image sharing scheme in which a secret
monochrome image is encrypted into two shares. Monochrome pixels have only two values either
black or white. The pixels of the two shares are determined with probability 50%. If the pixel is
white, one of the above two rows is chosen to generate two shares. Similarly If pixel is black, one

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International Journal of Network Security & Its Applications (IJNSA), Vol.6, No.4, July 2014
67
of the below two rows of is chosen to generate the two shares. Figure 1. shows the possible values
of pixels in each of the two generated shares.
Figure 1 : Naor-Shamir Scheme
The reconstruction of the secret image is simply done by stacking the pixels again as shown in
Figure 1. Although this scheme eliminates the overhead of complex computation present in
Shamir's scheme, however, the scheme is valid for monochrome images only and the secret image
can be detected and reconstructed by the human visual system with no much effort.
2.3. Thien-Lin ሺ࢑, ࢔ሻ Scheme
Thien and Lin [4] proposed a lossy ሺk, nሻ threshold image secret sharing scheme derived from
Shamir secret sharing method. The scheme generates small n shadows of size rxr from
polynomials of degree k െ 1. In this scheme, the pixels of greater than 250 are truncated and then
permuted with a secret key.
Each time k pixels are proccesed from the permuted set of pixels and a polynomial of degree
k െ 1 is created.
Fሺx୧ሻj ൌ p଴ ൅ pଵx୧ ൅ pଶx୧
ଶ
൅ ‫ڮ‬ ൅ pሺ୩ିଵሻx୧
ሺ୩ିଵሻ
mod 251, i ൌ 1,2, … , n (3)
Where m୧ are values from the set of permuted pixels and ݆ is the current iertation 1 ൑ j, ൑ r .The
݊ shares are calculated as F୨ሺ1ሻ, F୨ሺ2ሻ, Fሺ3ሻ, … , F୨ሺnሻ, and sequentially assigned to the n shadow
images Sଵ, Sଶ, Sଷ, … , S୬. The algorithm is repeated until all permuted pixels are processed. The
revealing process is implemented by Lagrange interpolation using any k shadow images. For each
j value, f୨ሺ1ሻ, f୨ሺ2ሻ, … , f୨ሺkሻ are computed to obtain the coefficients p଴, pଵ, pଶ, … , p୩ିଵ by
interpolation then the inverse of the permutation operation is applied to recover the original secret
image.
2.4. Quadratic Residues and Chen-Chang ሺ૛, ૛ሻ scheme
Let ‫݌‬ ߳ ࢆ and ‫݌‬ ߳ ࡺ . A number ‫ݔ‬ is called a quadratic residue (QR) modulo ‫݌‬ if ݃ܿ݀ሺ‫,ݔ‬ ‫݌‬ሻ ൌ 1
where gcd is the common greatest divisor for ‫ݔ‬ and ‫,݌‬ and the congruence
‫ݕ‬ଶ
‫ؠ‬ ‫ݔ‬ ݉‫݀݋‬ ‫݌‬ (4)
has a solution and x is called a quadratic nonresidue (QNR) ݉‫݋݈ݑ݀݋‬ ‫݌‬ if (4) has no solution.

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International Journal of Network Security & Its Applications (IJNSA), Vol.6, No.4, July 2014
69
‫ݎ‬ ൌ ‫ݎ‬௧ ‫ݐ‬ ‫א‬ ሼ1,2,3,4ሽ.The last phase is to split ‫ܯ‬ into two bytes ܽ௜ and ܾ௜,put each of it in a
shadow images ‫ܫ‬௔ and ‫ܫ‬௕ respectively.
The reconstruction phase:
Once the shadow images ‫ܫ‬௔ and ‫ܫ‬௕ are received, take each pixel from both images and
concatenate them to get the square root ‫ܯ‬Ԣ ሺ‫ܯ‬ᇱ
ൌ ܽ௜
ᇱ
ുܾ௜
ᇱ
ሻ .Now compute ‫݌‬௜
ᇱ
ൌ ‫ܯ‬ᇱଶ
݉‫݀݋‬ ݊ then
find the four square roots modulo ݊ for ‫݌‬௜
ᇱ
namely ‫ܯ‬ଵԢ, ‫ܯ‬ଶԢ, ‫ܯ‬ଷԢ ܽ݊݀ ‫ܯ‬ସԢ . Finally, choose the
correct parameter ‫ݎ‬௧ based on the corresponding ‫ܯ‬௧Ԣ where ‫ݐ‬ ‫א‬ ሼ1,2,3,4ሽ and get the original
image pixel by ‫݌‬௜ ൌ
௣೔
ᇲ
௥
݉‫݀݋‬ ݊.
3. THE PROPOSED SCHEME
In this section we propose a new ሺ݇, ݊ሻ secret image sharing scheme based on Chen-Chang
scheme and Shamir's method to encode an image S to n shadow images, ܵ1, ܵ2, ܵ3, … , ܵ݊. In a
way similar to [5], we use quadratic residues to encrypt image pixels, however, Chen-Chang's
scheme is a (2, 2) secret sharing scheme and the two shares are obtained and recovered by a
simple split and concatenation operation .If the two participants in the algorithm combine their
shares by the reveal algorithm, the secret is discovered . As a result, the sharing procedure used in
[5] is not suitable for the general ሺ݇, ݊ሻ threshold scheme. In this work, we extend the scheme by
Chen-Chang and replace the split and concatenation operation by using Shamir’s method of secret
sharing. The sharing and reveal steps of the proposed scheme are described below.
The Sharing Algorithm:
Input: the secret image S
Output: n shadow images
1- Decrease pixel value ‫݌‬௜ if it is greater than 238 to 238.
2- Shuffle the pixels of the original image S to obtain a permuted set of pixels O.
3- Choose the proper case for the pixel ‫݌‬௜ of O.
4- Choose the first element ‫ݎ‬ in a proper ܴ௜ where i ൌ 1,2,3,4.
5- Compute the product ‫݌‬పഥ ൌ ‫݌‬௜. ‫ݎ‬ .
6- Find the four square roots for ‫݌‬పഥ .
7- Choose ‫ܯ‬ ൌ ‫ܯ‬௜ according to ‫ݎ‬௜ .
8- Use Shamir secret sharing encoding to transform ‫ܯ‬ into ݊ shares.
9- Construct ݊ shadow images from the ݊ shares.
The Reveal Algorithm:
Input: ݇ ൏ൌ ݊ shadow images.
Output: the secret image S.
1. Perform the polynomial interpolation of Shamir decoding algorithm to get ‫ܯ‬Ԣ
2. Compute ‫݌‬పഥ ൌ ‫ܯ‬ᇱଶ
݉‫݀݋‬ ݊ .
3. Find the four square root of ‫݌‬పഥ .
4. Find appropriate parameter ‫ݎ‬ᇱ
according to the order of ‫ܯ‬Ԣ.
5. Compute ‫݌‬௜ ൌ ‫݌‬పഥ ‫ݎ‬Ԣ⁄ .
6. Put ‫݌‬௜ in the reconstructed image.
7. After all pixels ‫݌‬௜ are processed, Depermute the reconstructed O to get S.

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International Journal of Network Security & Its Applications (IJNSA), Vol.6, No.4, July 2014
71
In our experiment a (2,4) secret sharing scheme was chosen. i.e. Here n=4, k=2. Figure 3 shows
the secret image (Lena), the shares and the reconstructed image. Table 1 lists the experimental
result. In four cases, ∞ means that the proposed scheme give identical reconstructed image as the
original one and it is impossible to distinguish the secret image from the reconstructed one.
According to [11], the acceptable range of a lossy image is between 30 and 50 dB and only in one
case, the obtained value 0.93972 dB is much higher than the average.
Figure 3: The secret image, recovered image and the four noise-like image shares ܵ1 , … , ܵ4
5. CONCLUSION
In this paper a new ሺ݇, ݊ሻ threshold secret sharing method for grayscale images is introduced. The
concept of quadratic residues is used to encrypt the image and then Shamir's approach is
employed to produce the shadow images. The main drive of this work is to allow any number of
participants to share a secret image in a secure manner. The proposed scheme provides greater
flexibility and security in comparison with Chen-Chang's method. The experimental results shows
that advantages of our scheme are: (i) Identical or nearly similar reconstructed image from any k
shadow images, (ii) k shadow images with the same size as the original image. Our future work
will investigate the implementation of the proposed scheme effectively in embedded and mobile
devices.
Table 1. PSNR Values.
Secret Image Proposed Method
N=4, k=2
Peppers ∞
Lena 93.97 dB
Baboon ∞
Barbara ∞
Goldhill ∞